Problem: Simplify and expand the following expression: $ \dfrac{5}{k - 6}+ \dfrac{5}{k - 7}- \dfrac{2k}{k^2 - 13k + 42} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor the quadratic in the third term: $ \dfrac{2k}{k^2 - 13k + 42} = \dfrac{2k}{(k - 6)(k - 7)}$ Now we have: $ \dfrac{5}{k - 6}+ \dfrac{5}{k - 7}- \dfrac{2k}{(k - 6)(k - 7)} $ The least common multiple of the denominators is: $ (k - 6)(k - 7)$ In order to get the first term over $(k - 6)(k - 7)$ , multiply by $\dfrac{k - 7}{k - 7}$ $ \dfrac{5}{k - 6} \times \dfrac{k - 7}{k - 7} = \dfrac{5(k - 7)}{(k - 6)(k - 7)} $ In order to get the second term over $(k - 6)(k - 7)$ , multiply by $\dfrac{k - 6}{k - 6}$ $ \dfrac{5}{k - 7} \times \dfrac{k - 6}{k - 6} = \dfrac{5(k - 6)}{(k - 6)(k - 7)} $ Now we have: $ \dfrac{5(k - 7)}{(k - 6)(k - 7)} + \dfrac{5(k - 6)}{(k - 6)(k - 7)} - \dfrac{2k}{(k - 6)(k - 7)} $ $ = \dfrac{ 5(k - 7) + 5(k - 6) - 2k} {(k - 6)(k - 7)} $ Expand: $ = \dfrac{5k - 35 + 5k - 30 - 2k}{k^2 - 13k + 42} $ $ = \dfrac{8k - 65}{k^2 - 13k + 42}$